30
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
but the diffusion term still uses the central difference scheme of the second-order
truncation error.
2.3.2 Staggered Grid Method
The methods for establishing discrete equations include Taylor expansion method,
polynomial fitting method, control volume integral method, balance method, etc.;
and the three difference methods that are often used include forward difference,
backward difference, and center difference. The Taylor expansion method and the
polynomial fitting lay more stress on mathematical derivation, mainly by replacing
the derivatives in the governing equations with corresponding difference expressions
to form discrete equations. The control volume integral method and the balance
method lay particular stress on the physical point of view, and the physical meaning
is clear. Currently, popular commercial software such as PHOENICS, FLUENT,
STAR-CD, CFX, and FLOW-3D are developed based on the finite volume method.
There are two key problems in solving the nonlinear Navier-Stokers equation: one
is that the discrete form of momentum equation may fail to detect the unreasonable
pressure field when the pressure gradient term is discretized by using the conventional
grid and central difference method; the other is how to construct an equation for
calculating the pressure improvement value when the relationship between pressure
and velocity is implicit in the continuity equation. The two key problems mentioned
above are related to the dispersion of the pressure gradient and the solution of the
pressure, collectively referred to as the “coupling problem of pressure and velocity”.
If a waveform pressure field is obtained by a numerical solution, it is called decoupling
between pressure and velocity. To overcome the decoupling between pressure and
velocity, a staggered grid may be used.
The staggered grid means different grids are used for different variables. There
are three types of grids in the two-dimensional SIMPLE algorithm (semi-implicit
method for pressure-linked equations): the main control body grid, the control body
grid of x-direction velocity u and the control body grid of y-direction velocity v.
The main control body grid is suitable for pressure and scalar equations including
temperature and concentration. The control body for equations of velocity u and v is
dislocated from the main control body in spatial position.
Such a staggered grid for velocity component was first adopted by Harlow and
Welch in their MAC methods, and then applied to other methods developed by
Harlow and his colleagues, forming the basis of the SIVA program of Caretto, Curr
and Spalding and the basis of the SIMPLE program of Patankar and Spalding.
To solve the incompressible Navier-Stokers equation, the technique of the staggered grid is needed when the central difference method is used to discretize partial
differential equations.
A staggered grid has two important advantages: the staggered grid can avoid the
situation that the wave velocity field satisfies the continuity equation. The pressure
difference between two adjacent grid nodes becomes the natural driving force of the
2 Model of Quasi-Steady Weld Pool Dynamics and Numerical Simulation
but the diffusion term still uses the central difference scheme of the second-order
truncation error.
2.3.2 Staggered Grid Method
The methods for establishing discrete equations include Taylor expansion method,
polynomial fitting method, control volume integral method, balance method, etc.;
and the three difference methods that are often used include forward difference,
backward difference, and center difference. The Taylor expansion method and the
polynomial fitting lay more stress on mathematical derivation, mainly by replacing
the derivatives in the governing equations with corresponding difference expressions
to form discrete equations. The control volume integral method and the balance
method lay particular stress on the physical point of view, and the physical meaning
is clear. Currently, popular commercial software such as PHOENICS, FLUENT,
STAR-CD, CFX, and FLOW-3D are developed based on the finite volume method.
There are two key problems in solving the nonlinear Navier-Stokers equation: one
is that the discrete form of momentum equation may fail to detect the unreasonable
pressure field when the pressure gradient term is discretized by using the conventional
grid and central difference method; the other is how to construct an equation for
calculating the pressure improvement value when the relationship between pressure
and velocity is implicit in the continuity equation. The two key problems mentioned
above are related to the dispersion of the pressure gradient and the solution of the
pressure, collectively referred to as the “coupling problem of pressure and velocity”.
If a waveform pressure field is obtained by a numerical solution, it is called decoupling
between pressure and velocity. To overcome the decoupling between pressure and
velocity, a staggered grid may be used.
The staggered grid means different grids are used for different variables. There
are three types of grids in the two-dimensional SIMPLE algorithm (semi-implicit
method for pressure-linked equations): the main control body grid, the control body
grid of x-direction velocity u and the control body grid of y-direction velocity v.
The main control body grid is suitable for pressure and scalar equations including
temperature and concentration. The control body for equations of velocity u and v is
dislocated from the main control body in spatial position.
Such a staggered grid for velocity component was first adopted by Harlow and
Welch in their MAC methods, and then applied to other methods developed by
Harlow and his colleagues, forming the basis of the SIVA program of Caretto, Curr
and Spalding and the basis of the SIMPLE program of Patankar and Spalding.
To solve the incompressible Navier-Stokers equation, the technique of the staggered grid is needed when the central difference method is used to discretize partial
differential equations.
A staggered grid has two important advantages: the staggered grid can avoid the
situation that the wave velocity field satisfies the continuity equation. The pressure
difference between two adjacent grid nodes becomes the natural driving force of the